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Optimal lower and upper bounds for the $L_p$-mean deviation of functions of a random variable

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Bischoff, Wolfgang ; Fichter, M.:
Optimal lower and upper bounds for the $L_p$-mean deviation of functions of a random variable.
In: Mathematical methods of statistics. 9 (2000). - S. 237-269.
ISSN 1934-8045

Kurzfassung/Abstract

Lower and upper bounds for the variance (that is the mean deviation in $L_2$) of functions of a random variable are well-known in the literature where they are called Cacoullos, Chernoff, or Poincaré type inequality. A similar upper bound is known for the mean deviation of functions of a random variable in $L_1$. In the present paper we give a general approach to how one can obtain lower and upper bounds for the mean deviation of functions of a random variable in $L_p$, $1\le p<\infty$. Especially, we obtain each inequality of the above type stated in the literature. Moreover, we also state new inequalities. To be concrete, we demonstrate such new inequalities for the class of probability measures with Lebesgue density
$$f_p(x;\mu,\alpha)= c_p(\alpha)\cdot\exp(-|x-\mu|p/\alpha),\quad p\in [1,\infty).$$
Further, we develop lower and upper bounds for non-centered $p$th absolute moments of functions of a random variable, $p\in [1,\infty)$. For both types of inequalities we show that they are optimal in a certain sense. Our approach using Hille-Tamarkin operators is also new for the known results.

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Publikationsform:Artikel
Institutionen der Universität:Mathematisch-Geographische Fakultät > Mathematik > Lehrstuhl für Mathematik - Statistik und Stochastik
Peer-Review-Journal:Ja
Verlag:Allerton Press
Die Zeitschrift ist nachgewiesen in:
Titel an der KU entstanden:Nein
KU.edoc-ID:3709
Eingestellt am: 04. Mär 2010 13:37
Letzte Änderung: 04. Mär 2010 13:37
URL zu dieser Anzeige: https://edoc.ku.de/id/eprint/3709/
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